77.46.2 problem Ex 2 page 69

Internal problem ID [20851]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Book Solved Excercises. Chapter V. Singular solutions
Problem number : Ex 2 page 69
Date solved : Thursday, October 02, 2025 at 06:39:12 PM
CAS classification : [_quadrature]

\begin{align*} y^{2} \left (1+{y^{\prime }}^{2}\right )&=r^{2} \end{align*}

✓ Maple. Time used: 0.119 (sec). Leaf size: 54

ode:=y(x)^2*(1+diff(y(x),x)^2) = r^2; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -r \\ y &= r \\ y &= \sqrt {-c_1^{2}+2 c_1 x +r^{2}-x^{2}} \\ y &= -\sqrt {\left (r +x -c_1 \right ) \left (c_1 +r -x \right )} \\ \end{align*}

✓ Mathematica. Time used: 0.146 (sec). Leaf size: 101

ode=y[x]^2*(1+D[y[x],x]^2)==r^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\sqrt {r^2-(x+c_1){}^2}\\ y(x)&\to \sqrt {r^2-(x+c_1){}^2}\\ y(x)&\to -\sqrt {r^2-(x-c_1){}^2}\\ y(x)&\to \sqrt {r^2-(x-c_1){}^2}\\ y(x)&\to -r\\ y(x)&\to r \end{align*}

✓ Sympy. Time used: 2.699 (sec). Leaf size: 85

from sympy import * 
x = symbols("x") 
r = symbols("r") 
y = Function("y") 
ode = Eq(-r**2 + (Derivative(y(x), x)**2 + 1)*y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = - \sqrt {- C_{1}^{2} + 2 C_{1} x + r^{2} - x^{2}}, \ y{\left (x \right )} = \sqrt {- C_{1}^{2} + 2 C_{1} x + r^{2} - x^{2}}, \ y{\left (x \right )} = - \sqrt {- C_{1}^{2} - 2 C_{1} x + r^{2} - x^{2}}, \ y{\left (x \right )} = \sqrt {- C_{1}^{2} - 2 C_{1} x + r^{2} - x^{2}}\right ] \]